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In this article, we study multivariate polynomial interpolation andquadrature rules on non-tensor product node sets related to Lissajous curvesand Chebyshev varieties. After classifying multivariate Lissajous curves andthe interpolation nodes linked to these curves, we derive a discreteorthogonality structure on these node sets. Using this orthogonality structure,we obtain unique polynomial interpolation in appropriately defined spaces ofmultivariate Chebyshev polynomials. Our results generalize correspondinginterpolation and quadrature results for the Chebyshev-Gau{\ss}-Lobatto pointsin dimension one and the Padua points in dimension two.

Multivariate polynomial interpolation on Lissajous-Chebyshev nodes